A geometric characterisation of subvarieties of the standard E_6-variety related to the ternions, degenerate split quaternions and sextonions over arbitrary fields

TL;DR

This paper provides a geometric characterization of specific subvarieties within the E_6 variety, linking algebraic structures like sextonions and degenerate algebras to geometric objects over any field.

Contribution

It introduces a new geometric description of subvarieties related to sextonions and degenerate algebras within the E_6 variety, expanding understanding of their structure over arbitrary fields.

Findings

Characterization of subvarieties via Veronese representations

Connection to the Freudenthal-Tits magic square

Extension to degenerate composition algebras

Abstract

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety (the standard projective variety associated to the split exceptional group of Lie type E_6) over an arbitrary field K. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions over K (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal-Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square.